The International Arab Journal of Information Technology, Vol. 10, No. 4, July 2013
373
Design and Analysis of Array Weighted Wideband
Antenna using FRFT
1
Adari Satya Srinivasa Rao1 and Prudhivi Mallikarjuna Rao2
Department of ECE, Aditya Institute of Technology and Management, India
2
Department of ECE, Andhra University College of Engineering, India
Abstract: The beamwidth of a linear array depends on number of elements in the array and frequency of the input signal. The
main requirement of wideband beamformer is, the main beam pattern should be constant even there is a change in input signal
frequency. Various methods were proposed in literature, one method is called elemental lowpass filtering designed by using
Finite Impulse Response (FIR) digital filters. In this paper, the elemental lowpass filtering method was implemented using
Fractional Fourier Transform (FRFT) and performance analysis was carried out with array weighting.
.
Keywords: Antenna array, wideband antenna, FRFT, array weighting.
Received December 28, 2010; accepted May 24, 2011; published online August 5, 2012
1. Introduction
Arrays of broadband signals have been applied in
sonar, radio, radar, acoustic imaging etc., these are
often difficult to design because of highly frequency
dependent array properties. Figure 1 shows the
directivity pattern of a simple 21 element linear array.
The figure shows, the mainlobe width decreases with
increase in frequency. This causes, some signals to be
received with distorted spectra, and also frequencydependent null locations impair the ability to cancel
broadband interference.
21 elements antenna
0
14.844MHz
15.820MHz
16.796MHz
-5
X: -0.2696
Y: -13.35
Response in dB
-10
-15
-20
-25
-30
sidelobes can be suppressed by weighting, shading or
windowing the array elements. Array element
weighting has numerous applications in areas such as
digital signal processing, radio astronomy, radar, sonar,
and communications.
From Fractional Fourier Transform (FRFT) and
related concepts [1, 2, 8], it has been seen that the
properties and applications of Continuous Time
Fourier Transforms (CTFT) is special case of FRFT.
We have explained the usefulness of FRFT in the
design of broadband beamformers with uniform
spacing and advantages of it have been reported [7].
This paper gives the suitability and results of certain
array weighting methods in the design of broad-band
antenna array using FRFT. This paper is organized as:
section 2 presents theory for beamforming, section 3
presents implementation of lowpass filter with FRFT,
section 4 focused on some array weighting methods,
section 5 gives the simulation results and section 6
represents conclusions respectively.
-35
-40
-1
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
Figure 1. Plane wave response of a linear array with input
frequency.
In the past, broadband beamformers have been
studied extensively due to its applications [3, 4, 6]. It
was apparent that, for a uniformly weighted linear
array the largest sidelobes are down approximately 24
percent from peak value. The sidelobe levels can be
further decreased by using non uniform spacing
method. The presence of sidelobes means that the array
is radiating energy in untended directions. In a
multipath environment, the sidelobes can receive the
same signal from multiple angles. This is the basis for
fading experienced in communications [5]. The
2. Beamforming Theory
For a linear array, the far field response for an input
frequency ω and incident angle θ (measured relative to
broadside) is given by [6]:
P (θ , ω ) =
∞

∫ exp − j
−∞
ω sin θ 
c
x  D ( x , ω ) dx

(1)
where c and θ are the propagation speed and angle of
impinging signal and D(x,ω) the frequency response
with respect to the angular frequency ω and location x.
Obviously, in general p(ω,θ) is a function of both ω
and θ, while for a frequency invariant beamformer, we
require that the beam pattern p(ω,θ) be independent of
ω. For a weighted linear array of 2N+1 equally spaced
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The International Arab Journal of Information Technology, Vol. 10, No. 4, July 2013
unidirectional elements, the far field response can be
represented from the above equation 1:
P (θ , ω ) =
=
N
∑
n =−N
N
∑
ω sin θ 

ex p  − j
x  D (x , ω )
c


ex p ( − jn ω τ 0 sin θ ) D ( x , ω )
FRFT, which is controlled by a single continuous
angular parameter α. So, FRFT can be represented as
the rotation of signal in time-frequency plane [8] as
shown in Figure 3.
(2)
n =−N
where, τ0=d/c is the interelement spacing divided by
the speed of wave and D(x,ω) is the response of the
filter connected to antenna element at x. The inter-null
beamwidth of a uniformly excited array is given by
[4]:
 2π
θ BW = 2 sin −1 
 Mωτ 0

4π
 ≈
 Mωτ 0
(3)
Figure 3. Rotation of signal in time frequency plane with an angle α.
where, M=2N+1. This expression clearly indicates that
the beamwidth is inversely proportional to frequency
(or proportional to normalized frequency). It also
implies that an increase in either the number of
elements or interelement spacing results in a decrease
in the beamwidth as well.
Figure 2 presents the transmission signal of signal
from each antenna element and is passed through
lowpass filter. The weighted sum of lowpass filters are
combined at a linear combiner [3]. Here, we have used
FRFT to design lowpass filter.
The generalization of the CTFT is obtained if we
consider a rotation through an arbitrary angle α in the
(t, ω) plane. Thus, the FRFT of a signal f(t), can be
expressed as [7]:
∞
F a [ f ( t a ) ] = ∫ K a (t a ,t ) f ( t )dt
−∞
Ka (t a ,t ) = Kϕ exp  j π (t a2 cot ϕ − 2 t a t cos ecϕ + t 2 cot ϕ ) 
K φ = exp [− j (π sgn (φ ) / 4 − φ / 2 )] [sin φ ]0 . 5
(5)
(6)
where φ=aπ/2.
The kernel function Ka(ta,t) has the following spectral
expansion:
K a (t a , t ) =
∞


∑ψ k (t a ) exp −
k =0
j
π
2

ka ψ k (t )

(7)
where ψk(t) denotes kth Hermite-Gaussian function,
and ta denotes the variable in the ath-order Fractional
Fourier Domain. The kth order Hermite-Gaussian
function is defined as (k=0,1,2,…):
ψ k (t ) =
Figure 2. Filter-and-sum structure.
3. Implementation of Low Pass Filter using
FRFT
The CTFT, which is defined by the following pair [8]:
F (ω ) =
f
(t ) =
1
2π
1
2π
∞
∫f
−∞
∞
( t ) ex p ( − j ω t ) dt ↔
∫ F ( ω ) ex p ( j ω t ) d ω
(4)
−∞
The CTFT reflects to the assumption that the signal of
interest has stationary frequency content. However,
signal representations using intermediate “angularly
coupled axes” hold some promise for analyzing signals
with time-frequency coupling, e.g., linear-frequency
modulation. Angular transform of the CTFT, which is
called as the Angular Fourier Transforms (AFT) or
21 4
2 k k!
Hk
(
) (
2 π t exp − π t 2
)
(8)
where Hk denotes kth order Hermite polynomial having
π 

k real zeros. In the equation 7, exp − j 2 ka  represents


the ath power of the eigenvalues. When a=1, the FRFT
reduces to the ordinary fourier transform, where t1
denotes the frequency-domain variable. Here, we have
adopted the design low pass filter using FRFT based
on procedure of tunable Finite Impulse Response (FIR)
filters. A FIR digital filter operation is a linear
convolution of the finite duration impulse response
with the input signal sequence x(n). The impulse
response h(n), of kaiser window is given as [9]:
h(n) = hd (n) w( n)
(9)
where hd(n) is the desired or ideal impulse response,
and w(n) is the kaiser window sequence. Since
multiplication in the time domain corresponds to
Design and Analysis of Array Weighted Wideband Antenna using FRFT
convolution in the frequency domain which can be
expressed as a complex convolution operation given
as:
H (ω ) =
1
2π
π
∫ W (λ )H d (ω
− λ )d λ
375
4.3. Hamming Weights
The hamming weights are defined by:
w( k + 1 ) = 0.54 − 0.46 cos( 2πk /( N − 1 ))
(10)
−π
(12)
where k=0, 1,…, N-1.
where, H(ω) is the frequency response of filter Hd(ω),
is desired or ideal frequency response of lowpass filter,
and W(ω) is frequency response of kaiser window.
From the above equation, the transition bandwidth of is
proportional to mainlobe width of kaiser window
W(ω). Figure 4 shows the variation of mainlobe width
with order of FRFT. It is observed that as the FRFT
order is reduced the main lobe width of FRFT kaiser
window shrinks. This feature has been used here to
tune the transition bandwidth of lowpass filter.
4.4. Kaiser-Bessel Weights
The kaiser-bessel weights are defined by:
w( k ) =

2
 k  

I 0 πα 1 − 

λ
2

 


(13)
I 0 [πα ]
where k=0, 1,…, N-1 and α>1.
4.5. Blackman-Harris Weights
10
a1=0.2
a2=0.4
a3=0.6
0
The blackman-harris weights for -92dB sidelobe level
are defined by:
Response in dB
-10
2 π k 
w ( k + 1 ) = 0 .3 5 8 7 5 − 0 .4 8 8 2 9 c o s 

N −1 
4 π k 
6 π k 
+ 0 .1 4 1 2 8 c o s 
 . − 0 .0 1 1 6 8 c o s  N − 1 
N −1 


-20
-30
-40
(14)
-50
-60
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized frequency
0.7
0.8
0.9
4.6. Nuttall Weights
1
Figure 4. Variation in kaiser window response with order of FRFT.
The Nuttall weights are defined by:
2π k 
w ( k + 1 ) = 0.35576 − 0.48739 cos 

 N −1 
4 π k 
6 π k 
+ 0.14423 cos 
 . − 0.01260 cos  N − 1 
 N −1 


4. Array Weighting Methods
There are a vast number of possible window functions
available that can provide weights for use with linear
arrays. Some of the more common window functions
are considered for analysis of wideband antenna.
4.1. Binomial Weights
Binomial weights will create an array factor with
sidelobes, provided that the element spacing d ≤ λ
The binomial weights are chosen from the rows
Pascal’s triangle. The first five rows are shown
Table 1.
no
2.
of
in
Table 1. Pascal’s triangle.
5. Simulation Results
The design results were evaluated using computer
simulation, for frequencies 14.844MHz, 15.820MHz,
and 16.796MHz, for 21 element antenna array with
uniform spacing. Figures 5, 6, 7, 8, 9 and 10 show the
antenna pattern and the frequency characteristics of the
wideband antenna with FIR Low Pass filter and FRFT
Low Pass Filters for Binomial, Gaussian, Hamming,
Kaiser-Bessel, Blackman-Harris and Nuttall weighting
methods.
1
1
1
1
21 elements wideband antenna with Binomial weighting
2
3
6
0
14.844MHz
15.820MHz
16.796MHz
-20
1
3
4
21 elements wideband antenna with Binomial weighting - FRFT
0
1
-40
-20
-60
1
4
14.844MHz
15.820MHz
16.796MHz
-10
-30
-80
1
|AF| in dB
1
|AF| in dB
N=1
N=2
N=3
N=4
N=5
(15)
-100
-120
-40
-50
-140
-60
4.2. Gaussian Weights
-160
-70
-180
The gaussian weights are defined by:
w( k + 1 ) = e
2
1 k −N 2 

−  α
2
N 2 
where k=0, 1,…, N and α≥2.
-200
-1
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
a) FIR low pass filter.
(11)
0.8
1
-80
-1
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
Figure 5. Response of 21-element wideband antenna array designed
with binomial weights.
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The International Arab Journal of Information Technology, Vol. 10, No. 4, July 2013
21 elements wideband antenna with Gaussian weighting
21 elements wideband antenna with Gaussian weighting - FRFT
0
-10
14.844MHz
15.820MHz
16.796MHz
-10
-20
-20
-30
|AF| in dB
|AF| in dB
6. Conclusions
0
14.844MHz
15.820MHz
16.796MHz
-30
-40
-40
-50
-50
-60
-60
-1
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
-70
-1
1
-0.8
a) FIR low pass filter.
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
Figure 6. Response of 21-element wideband antenna array designed
with Gaussian weights.
21 elements wideband antenna with Hamming weighting
21 elements wideband antenna with Hamming weighting - FRFT
0
0
14.844MHz
15.820MHz
16.796MHz
-40
-40
-60
-60
-80
-100
References
[1]
-80
-100
-120
-140
-1
14.844MHz
15.820MHz
16.796MHz
-20
|AF| in dB
|AF| in dB
-20
In this paper we have discussed about application of
array weighting on the design of wideband antennas.
From the Figures 5, 6, 7, 8, 9 and 10, it is evident that
if the low pass filters are implemented by using FRFT,
gives better invariant radiation patterns compared to
normal FIR low pass filters. Figures 5, 9 and 10 shows
that the Binomial weights, Blackman-Harris weights
and Nuttall weights gives lower side lobe levels, more
than 80dB with FIR filters and more than 60dB with
FRFT filters. From Figure 8 it is evident that kaiserbessel weights gives better directional characteristics.
-120
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
-140
-1
1
-0.8
a) FIR low pass filter.
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
[2]
Figure 7. Response of 21-element wideband antenna array designed
with Hamming weights and FRFT Low Pass Filter.
21 elements wideband antenna with Kaiser - Bessel weighting - FRFT
21 elements wideband antenna with Kaiser - Bessel weighting
0
0
14.844MHz
15.820MHz
16.796MHz
-20
[3]
14.844MHz
15.820MHz
16.796MHz
-20
-40
|AF| in dB
|AF| in dB
-40
-60
-60
-80
-80
-100
-100
-120
-1
[4]
-120
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
-140
-1
1
-0.8
a) FIR low pass filter.
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
Figure 8. Response of 21-element wideband antenna array designed
with Kaiser-Bessel weights.
[5]
21 elements wideband antenna with Blackman - Harris weighting
21 elements wideband antenna with Blackman - Harris weighting - FRFT
0
0
14.844MHz
15.820MHz
16.796MHz
-20
-60
-30
-80
-50
-120
-60
-140
-70
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
-80
-1
1
[6]
-40
-100
-160
-1
14.844MHz
15.820MHz
16.796MHz
-10
-40
|AF| in dB
|AF| in dB
-20
-0.8
a) FIR low pass filter.
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
[7]
Figure 9. Response of 21-element wideband antenna array designed
with Blackman-Harris weights.
21 elements wideband antenna with Nuttall weighting
21 elements wideband antenna with Nuttall weighting - FRFT
0
0
14.844MHz
15.820MHz
16.796MHz
-20
14.844MHz
15.820MHz
16.796MHz
-10
-40
-20
-60
[8]
|AF| in dB
|AF| in dB
-30
-80
-100
-50
-120
-60
-140
-70
-160
-180
-1
-40
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
a) FIR low pass filter.
0.8
1
-80
-1
-0.8
-0.6
-0.4
-0.2
0
U
0.2
0.4
0.6
0.8
1
b) FRFT low pass filter.
Figure 10. Response of 21-element wideband antenna array
designed with Nuttall weights.
[9]
Almeida L., “The Fractional Fourier Transform
and Time-Frequency Representation,” IEEE
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Design and Analysis of Array Weighted Wideband Antenna using FRFT
Adari Satya Srinivasa Rao
received his M Tech. degree from
Andhra University, Vishakapatnam
in 2004 and he is a PhD student in
Andhra University. He is having 15
years of teaching experience in
various
engineering
colleges.
Presently, he is working as Professor, Department of
ECE, Aditya Institute of Technology and Management,
Tekkali. His interests include signal processing,
adaptive antenna arrays and communication systems.
377
Prudhivi
Mallikarjuna
Rao
received his ME degree from
Andhra University in 1985 and PhD
degree from Andhra University in
1998. He is now working as
Professor, Department of ECE,
AUCE, Vishakaptnam. His interests
include EMI/EMC, antenna array syntesis and applied
electromagnetic applications. He has published 25
papers in various journals/proceedings in the field of
antenna arrays and EMI/EMC.